Integrand size = 15, antiderivative size = 270 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {7 b \sqrt {a+\frac {b}{x^3}} x^2}{20 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^5}{5 a}-\frac {7 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} a^2 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:
-7/20*b*(a+b/x^3)^(1/2)*x^2/a^2+1/5*(a+b/x^3)^(1/2)*x^5/a-7/60*(1/2*6^(1/2 )+1/2*2^(1/2))*b^(5/3)*(a^(1/3)+b^(1/3)/x)*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b ^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^(1/2)*EllipticF(((1-3^(1/2))* a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x),I*3^(1/2)+2*I)*3^(3/4)/ a^2/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1 /3)/x)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {-7 b^2-3 a b x^3+4 a^2 x^6+7 b^2 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{20 a^2 \sqrt {a+\frac {b}{x^3}} x} \] Input:
Integrate[x^4/Sqrt[a + b/x^3],x]
Output:
(-7*b^2 - 3*a*b*x^3 + 4*a^2*x^6 + 7*b^2*Sqrt[1 + (a*x^3)/b]*Hypergeometric 2F1[1/6, 1/2, 7/6, -((a*x^3)/b)])/(20*a^2*Sqrt[a + b/x^3]*x)
Time = 0.51 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {858, 847, 847, 759}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle -\int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {7 b \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{10 a}+\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {7 b \left (-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )}{10 a}+\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {7 b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )}{10 a}+\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\) |
Input:
Int[x^4/Sqrt[a + b/x^3],x]
Output:
(Sqrt[a + b/x^3]*x^5)/(5*a) + (7*b*(-1/2*(Sqrt[a + b/x^3]*x^2)/a - (Sqrt[2 + Sqrt[3]]*b^(2/3)*(a^(1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a ^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[ArcSin [((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*a*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^ (1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])))/(10*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (207 ) = 414\).
Time = 1.30 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.75
| method | result | size |
| risch | \(\frac {\left (4 a \,x^{3}-7 b \right ) \left (a \,x^{3}+b \right )}{20 a^{2} x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}+\frac {7 b^{2} \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, {\left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}^{2} \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{\left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right ) \sqrt {x \left (a \,x^{3}+b \right )}}{20 a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {a x \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}\, x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) | \(742\) |
| default | \(\text {Expression too large to display}\) | \(2017\) |
Input:
int(x^4/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/20*(4*a*x^3-7*b)/a^2/x*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)+7/20*b^2/a*(1/2/a *(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*((-3/2/a*(-a^2*b)^(1/3)+1/ 2*I*3^(1/2)/a*(-a^2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a ^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(x-1/a*(-a^2*b)^(1/3))^2*(1/a*( -a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(-1/ 2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3))) ^(1/2)*(1/a*(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b )^(1/3))/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a ^2*b)^(1/3)))^(1/2)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)) /(-a^2*b)^(1/3)/(a*x*(x-1/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I* 3^(1/2)/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b) ^(1/3)))^(1/2)*EllipticF(((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^ (1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a ^2*b)^(1/3)))^(1/2),((3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)) *(1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(1/2/a*(-a^2*b)^(1/ 3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(3/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*( -a^2*b)^(1/3)))^(1/2))/x^2/((a*x^3+b)/x^3)^(1/2)*(x*(a*x^3+b))^(1/2)
\[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{4}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \] Input:
integrate(x^4/(a+b/x^3)^(1/2),x, algorithm="fricas")
Output:
integral(x^7*sqrt((a*x^3 + b)/x^3)/(a*x^3 + b), x)
Time = 0.69 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.17 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x^{5} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {2}{3}\right )} \] Input:
integrate(x**4/(a+b/x**3)**(1/2),x)
Output:
-x**5*gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), b*exp_polar(I*pi)/(a*x**3))/ (3*sqrt(a)*gamma(-2/3))
\[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{4}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \] Input:
integrate(x^4/(a+b/x^3)^(1/2),x, algorithm="maxima")
Output:
integrate(x^4/sqrt(a + b/x^3), x)
\[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{4}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \] Input:
integrate(x^4/(a+b/x^3)^(1/2),x, algorithm="giac")
Output:
integrate(x^4/sqrt(a + b/x^3), x)
Timed out. \[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \,d x \] Input:
int(x^4/(a + b/x^3)^(1/2),x)
Output:
int(x^4/(a + b/x^3)^(1/2), x)
\[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {8 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a \,x^{3}-14 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, b +7 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a \,x^{4}+b x}d x \right ) b^{2}}{40 a^{2}} \] Input:
int(x^4/(a+b/x^3)^(1/2),x)
Output:
(8*sqrt(x)*sqrt(a*x**3 + b)*a*x**3 - 14*sqrt(x)*sqrt(a*x**3 + b)*b + 7*int ((sqrt(x)*sqrt(a*x**3 + b))/(a*x**4 + b*x),x)*b**2)/(40*a**2)